Highly accurate simulation
of electrokinetic transport

Tinh Vo, Dr-Ing. Holger Marschall

Computational Multiphase Flow, TU Darmstadt

19 Oct, 2025

Progress Overview

Progress Failure
WP1 Implementation of classical Poisson-Nernst-Planck electrokinetic transport
- Segregated solver
- Numerical stability
Parametric Case Setup:
- Pore geometry coupling with electrochemical dimensionless numbers
Late times charging dynamics verification
- Large computational cost
WP2 Zero-flux boundary layer
- Sub-grid zero-flux boundary layer
Mass conservation testing
- Benchmark incompatibility
WP3 Extensible class inheritance structure
- Modifications of Poisson-Nernst-Planck transport equations available
WP4
WP5

Numerical Methods

Algorithm

Navier-Stokes Equations

\[ \frac{\partial}{\partial t}[\rho\mathbf{v}] + \nabla \cdot \{\rho\mathbf{v}\mathbf{v}\} = -\nabla p + \nabla \cdot \left\{\mu\left[\nabla\mathbf{v} + (\nabla\mathbf{v})^{\text{T}}\right]\right\} \]

\[ \frac{\partial\rho}{\partial t} +\nabla\cdot (\rho\mathbf{u}_{}) = 0 \]

Species Transport &

Gauss law

\[ \frac{ \partial c_{i} }{ \partial t } = \nabla \cdot(D_{i}\nabla c_{i}) + \nabla \cdot [( \underbrace{ D_{i} \dfrac{ez_{i}}{kT} } _{ \text{Stokes-Einstein relation} }\nabla \Psi)c_{i} ] \]

\[ \nabla^2 \Psi _{E} =\frac{-\rho_E}{\epsilon\epsilon_{0}} = \frac{-F \sum_{i} z_{i} c_{i}}{ \epsilon\epsilon_{0}} \]

Algorithm:

Coupling Algorithm

Boundary conditions

  • Zero Ionic Flux
    Neumann type
    • Enforcing zero species concentration flux across the charged wall.

\[ \left[ \nabla \cdot c_{i} \right]_{f} \cdot n_{\perp} \mathbf{A}_{_{f}} = - \left[ \frac{\mu_{i}}{D_{i}}\left( \nabla \Psi \right)c_{i} \right]_{f} \cdot \mathbf{A}_{_{f}} \]

\[ \longrightarrow \nabla c_{i,P} + \frac{\mu_{i}}{D_{i}}\left( \nabla \Psi_{f} \right)c_{i,P} = 0 \]

  • Subgrid Zero Ionic Flux
    Dirichtlet-type
    • Enforces zero species flux.
    • Includes exponential concentration distribution in the cells next to charged pore wall.

$$

c_{i,P}^P = c_{i,f}^S ( ( { {f} }
-
{
{P} }
) )

$$

Not tested!

Implemented, but incompatible with benchmark described by 1

Results

Electric potential & species transport

Case Setup1
  • Results for \(D = 1e-9 \mathrm{m/s^2},\frac{\lambda_D}{H}= 20.7 \mathrm{m}\ ,c_{Bulk} = 0.001 \mathrm{M}, \Psi_{wall} = 0.025 \frac{k_BT}{e}, t = 0.036 \frac{4}{\pi{}^{2}} \frac{L^{2}}{D} \frac{2\lambda{}_{D }}{{H}} \cosh \left( \frac{\Psi_{wall}}{2 \Psi_T} \right)\)
    • Corresponds to 8600000 time steps for 9500 cells!

Electric potential field at pore inlet

Salt concentation field at pore inlet

Charging dynamics

Analytical charging rate of a uniformly charged 2D nanopore 1

Outlook

  • Reference solutions
    • Early-time dynamics1: Instantaneous concentration and electric potential field step response in nanopore
    • Late-time (charging) dynamics2: Bi-exponental slowdown of species concentration accumulation
  • Improvement candidates:

◦ Block matrix coupling scheme 3 (WP1)
Stability and cost

◦ Modified Poisson-Nernst-Planck variations (WP3)
Mass Conservation, transport dynamics near walls, moderate voltages

Fuhrmann et al.4’s proposal to modify the Gouy-Chapman solution near walls for moderate voltages